Question

Two wires of same length and radius are joined in parallel and loaded if youngs modulus of wires are y1 and y2

Answer 1

Originally Answered: Two rods having Young's modulus Y1 and Y2 are attached in a parallel combination. If their length and cross-sectional area are same, what will be the total of Y?

Thanks for the A2A !
The total Y is going to be this,
Y=Y1+Y22Y=Y1+Y22
As given, the length and cross-sectional area of the two rods are same. Let’s assume they are L and A respectively. Now after combining them parallel, the total cross sectional Area will become 2A, the length is going to stay the same I mean L. Now if we apply force F on this system, both of the rods will not feel the same Force F. Rather the force will divide into two parts, which are, let’s assume F1F1and F2F2for the rods of young’s modulus Y1Y1and Y2Y2respectively.
It’s clear that F=F1+F2F=F1+F2
and the increase in length l is also the same for both of the rods.
From hooke’s law, we can write,
Y1=F1LAlY1=F1LAl
Y2=F2LAlY2=F2LAl
Let, the total Young’s Modulus of the combination is Y
Considering the whole Combination as a single rod we can write using hooke’s law,
Y=FL2AlY=FL2Al
=(F1+F2)L2Al=(F1+F2)L2Al
=F1L2Al+F2L2Al=F1L2Al+F2L2Al
=Y12